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Thread: Infinite Limit Proof

  1. #1
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    Infinite Limit Proof

    I need to prove:

    lim (x^4)-(2(x^2)) = +inf
    x->+inf

    Any suggestions?
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi,

    Hint : $\displaystyle x^4-2x^2=x^4\left(1-\frac{2}{x^2}\right)$
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  3. #3
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    I don't know how to do it. The problem is done and over with, but I still would like to know how to get the answer for my own knowledge.
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Calc1stu View Post
    I don't know how to do it. The problem is done and over with, but I still would like to know how to get the answer for my own knowledge.
    $\displaystyle \lim_{x\to\infty}x^4\left(1-\frac{2}{x^2}\right)=\lim_{x\to\infty}x^4\cdot\lim _{x\to\infty}\left(1-\frac{2}{x^2}\right)=\infty\cdot{1}=\infty$
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  5. #5
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    It was supposed to be an epsilon delta proof.
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  6. #6
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    You want to show that for every $\displaystyle M > 0$, there exists $\displaystyle N > 0$ such that whenever $\displaystyle x > N$, it necessarily follows that $\displaystyle f(x) > M \Leftrightarrow x^4 - 2x^2 > M $

    Consider the latter expression as an equality:
    $\displaystyle x^4 - 2x^2 = M $

    $\displaystyle \Leftrightarrow x^4 - 2x^2 - M = 0$

    $\displaystyle \Leftrightarrow x^2 = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-M)}}{2}$

    $\displaystyle \Leftrightarrow x^2 = \frac{2 \pm \sqrt{4 + 4M}}{2}$

    $\displaystyle \Leftrightarrow x^2 = 1 \pm \sqrt{1 + M} $

    $\displaystyle \Leftrightarrow x^2 = 1 + \sqrt{1 + M}$ (since $\displaystyle x^2 \geq 0$ and $\displaystyle M > 0$)

    $\displaystyle \Leftrightarrow x = +\sqrt{1 + \sqrt{1 + M}}$ (we're considering positive x's)

    So when $\displaystyle x \ {\color{red}=} \ \sqrt{1 + \sqrt{1 + M}}$ we have that $\displaystyle f(x)\ {\color{red} =}\ M$. We want to show that for some $\displaystyle x \ {\color{magenta}>}\ N$, we have that $\displaystyle f(x) \ {\color{magenta}>}\ M$

    Now consider: $\displaystyle x \ {\color{magenta}> }\ \sqrt{1 + \sqrt{1 + M}} > 0$. Can you finish?
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  7. #7
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    As far as epsilon and delta are concerned which is M and N. I don't know where to go from there. All we ever usually do or did was a basic function.
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